where the min and max are taken after all \(SG(d)\)  cocharacters \(\lambda \) , see [1]} applied to the \(T(d)\) -representation \(R(d)\) , for \(\varepsilon =0\) , and for the (slighly) larger polytope \(\mathbb {W}\)  considered in this paper. When \(\chi \in M^+_{\mathbb {R}}\) , we have that:
\(r=-\text{min}\,\frac{\langle \lambda , \chi \rangle }{\langle \lambda , N^{\lambda >0}\rangle },\) 
